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| Platonov V., Rapinchuk A. — Algebraic groups and number theory |
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| Предметный указатель |
(Sylow) pro-q-subgroup 139
Absolute rank 57
Absolutely simple group 62
Actions of algebraic groups on varieties 99
Additive Jordan decomposition 52
Adele group 249
Adele group, cohomology of 297
Adele group, finite 250
Adele group, integral 244 249
Adele group, principal 249
Adele group, principal 5- 250
Adele group, S- 249
Adele group, S-integral 249
Adele group, T-integral S- 250
Adele ring 10
Adele ring, 5-integral 11
Adele ring, integral 11
Adele ring, principal 11
Adele space 243
Adele space, principal 244
Adele space, S-integral 244
Adele topology 11
Adeles 10 243
Adelization of a regular K-map 244
Adjoint representation 51
Algebraic group 47
Algebraic group, adjoint 62
Algebraic group, character of 52
Algebraic group, classical simply connected simple algebraic K-groups, list of 92
Algebraic group, classification of K-forms of 75
Algebraic group, diagonalizable 52
Algebraic group, K-split 52 57 58
Algebraic group, quasisplit 58
Algebraic group, reductive 58
Algebraic group, semisimple 58
Algebraic group, simple 62
Algebraic group, simply connected 62
Algebraic group, unipotent 56
Algebraic number field 1
Algebraic tori 52
Analytic structure 110
Anisotropic groups, K- 65
Anisotropic kernel 67
Anisotropic torus, K- 53
Approximation, absolute strong 250
Approximation, strong 13 250 399
Approximation, strong, in groups, criterion for 427
Approximation, strong, theorem 427
Approximation, strong, theorem for a field 14
Approximation, weak 13 399
Approximation, weak, in algebraic groups 415
Approximation, weak, theorem for a field 14
Arithmetic subgroup 171
Arithmetic subgroup, O- 227
Arithmetic subgroup, S- 175 267 268
Arithmetic topology 555
Arithmeticity, conjecture on 437
B-N pair 149
Bartels — Kitaoka theorem 234
Bartels’ problem 309
Bartels’ theorem 492
Behr’s theorem 152
Birational isomorphism 98
Borel — Harish — Chandra theorem 193
Borel, measure 159
Borel, subgroups 57
Brauer group 28
Brauer — Hasse — Noether theorem 38
Bruhat decomposition 60
Building 149
Cartan decomposition 150
Cayley — Dickson parametrization 403
Central isogeny 62
Centralizer theorem 448
Chebotarev density theorem 9
Chernousov’s theorem 387
Chevalley base 64
Chevalley group 65
Chevalley’s theorem 99
Chevalley’s theorem, stronger version 100
class 444 449
Class group 452
Class number 2 251 440
Class number, in the genus 444 448 449
Class number, in the genus, of a quadratic form 447
Class number, in the genus, of an element 448
Class number, in the genus, of lattices in the full matrix algebra under conjugation 465
Closed Orbit Lemma 99
Coboundary map 23
Cohomological dimension 340
Cohomology 16
Cohomology, continuous 20
Cohomology, Galois 21
Cohomology, non-Abelian 21
Cohomology, of adele groups 297
Cohomology, of algebraic groups 71
Cohomology, of groups of w-adic integral points 292
Cohomology, real 320
Cohomology, Tate 300
Cohomology, unramified 294
Commensurability subgroup 206
Compact type 205
Compactly presented group 152
Compactness, criterion for , of  207
Compactness, criterion for of  260
Completion 3
Congruence, kernel 555
Congruence, subgroup 31 134 172 553
Congruence, subgroup conjecture (of Serre) 556
Congruence, subgroup problem 553
Congruence, theorem 525
Congruence, topology 555
Connected component 51
Continuous cohomology 20
Convergence coefficients 261
Corestriction map 20
Decomposition group 6
Decomposition, Bruhat 60
Decomposition, Cartan 150
Decomposition, Iwasawa, for  129
Decomposition, Iwasawa, theorem 131
Decomposition, Jordan 52
Decomposition, Jordan, additive 52
Decomposition, Levi 58
Decomposition, polar 124 126
Degree of an algebraic group 48
Degree, residue 5
Density Theorem, Borel 205
Density Theorem, Chebotarev 9
Diagonalizable algebraic group 52
Dieudonne’ determinant 39
Differential form 165
Differential form, integrating 165
Differential form, invariant 165
Dirichlet unit theorem 209
Distinguished vertex 66
Dominant morphism 96
E.Cartan’s theorem 407
Eichler’s theorem 38
Eichler’s theorem, unitary version 360
Eisenstein polynomial 9
Exponential map 116
Exponential map, truncated 56
Extension, K-adequate 309
Extension, totally ramified 5
Extension, totally real 229
Extension, unramified 5
Field, algebraic number 1
Field, local 5
Field, local, non-archimedean 5
| Field, of definition of an algebraic variety 96
Field, of type (F) 316
Field, residue 5
Field, splitting 27 53
Field, Weak approximation theorem for a 14
Finite presentability of S-arithmetic groups 272
Finiteness theorem for the orbits of arithmetic groups 193
Forms, differential 165
Forms, inner 66
Forms, integrating differential 165
Forms, invariant differential 165
Forms, K-forms of algebraic groups, classification of 75
Forms, Killing 51
Forms, L/K- 67
Forms, outer 66
Fractional ideals 2
Frobenius automorphism 6
Fundamental domain 163
Fundamental group 63
Fundamental set 164 189 193 253 267
Galois cohomology 21
Generalized Dirichlet’s theorem 276
Generalized Siegel set 224
Genus 444 449
Genus problem 494
Genus problem, for integral representations 504
Genus problem, in arithmetic groups 494
Grothendieck’s problem 434
Group, absolutely simple 62
Group, adele see “Adele group”
Group, algebraic 47
Group, algebraic, haracter of 52
Group, classification of semisimple groups 63
Group, compactly presented 152
Group, conjugacy separable 502
Group, decomposition 6
Group, defined over K see “Group K-”
Group, ideal class 1 2
Group, inertia 10
Group, isogeneous 56
Group, K- 49
Group, K-anisotropic 65
Group, K-isotropic 65
Group, of cocharacters see “One-parameter subgroups”
Group, of compact type 228
Group, of mixed type 228
Group, of noncompact type 228
Group, of S-units 267
Group, of type (F) 316
Group, of units 171
Group, orthogonal 80
Group, pro-p- 138
Group, profinite 137
Group, ramification 10
Group, reductive 58
Group, self-adjoint 124
Group, semisimple 58
Group, simple 62
Group, special linear 78
Group, special unitary 84
Group, spinor 82
Group, symplectic 80
Group, unimodular 160
Group, unitary 84
Group, with bounded generation 203
Haar measure 159
Haar measure, on a finite direct product 160
Harish — Chandra’s theorem 183
Hasse norm principle 15 307
Hasse norm theorem 308
Hasse principle 284 285
Hasse principle, first obstruction to 309
Hasse principle, for cohomology of arithmetic subgrou simply connected groups 491
Hasse principle, for simply connected groups 286
Hasse principle, for tori 307
Hasse principle, strong 347
Hasse principle, total obstruction to 311
Hasse principle, weak 347
Hasse — Witt invariant 349
Hensel’s Lemma 143
Hermite’s theorem 9
Hilbert class field 443
Hilbert’s Theorem 90 70
Hochschild — Serre spectral sequence 19
Hochschild — Serre spectral sequence, for non-Abelian cohomology 25
Ideal class group 1 2
Idele topology 12
Ideles 10
Ideles, group of 11
Ideles, group of, integral 12
Ideles, group of, principal 12
Ideles, group of, S-integral 12
Ideles, group of, special 12
INDEX 27
Inertia group 10
Inflation map 19
Inner form 66
Invariant, differential form 165
Invariant, Hasse — Witt 349
Invariant, measure on a quotient space 162
Invariant, of a division algebra 29
Invariant, of a simple algebra 29
Inverse function theorem 110
Inverse limit 137
Inverse system 137
Involution 83
Involution, of the first kind 83
Involution, of the first type 85
Involution, of the second kind 83
Involution, of the second type 85
Isogeny 56 62
Iwahori subgroup 148
Iwasawa decomposition for 129
Iwasawa decomposition theorem 131
Jordan decomposition 52
Killing form 51
Kneser — Tits conjecture 406
Kneser’s theorem 462
Krasner’s lemma, unitary version 364
Landherr’s theorem 359
Lang’s isogeny theorem 290
Lang’s theorem 281
Lattice (on a vector space) 42
Lattice (on a vector space), criterion for free 442
Lattice (on a vector space), in a locally compact topological group 221
Lattice (on a vector space), local 42
Levi decomposition 58
Lie algebra of an algebraic group 51
Lie algebra of an analytic group 116
Lie subgroup 117
Local field 5
Local field, non-archimedean 5
Local lattice 42
Local-global principle 14 285
Logarithmic map 116
Logarithmic map, truncated 56
Mahler’s criterion 211
Margulis’ conjecture 511
Margulis’ theorem 517
Matsumoto’s theorem 139
Measure 159
Metaplectic conjecture, weak 532 560
Metaplectic kernel 557
Meyer’s theorem 342
Minkowski — Hasse Theorem 14
Minkowski’s lemma 232 493
Mixed type 205
Module of a group 160
Module of an automorphism 159
Morphism of algebraic groups 48
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